The Taylor and other methods of summability

Abstract

are regular if and only if 0 <q <1 as proved in [3] and [4]. The question of inclusion of T(r) and S(q) was discussed by Schoonmaker [4]. In ??2 and 3 it will be shown that both T(r) and S(q) include the binary method Y, [5], and that neither T(r) nor S(q) includes the Cesaro method C(1) or the Abel method A. ?4 consists of the determination of domains in which the geometric series Zn and the power series EanZn are summable by the iterated product method B* T(r), where B denotes the Borel exponential method. A summary of similar results for the product method B S(q) comprises ?5. The author is indebted to his referee for his several helpful comments and especially for his suggestions which led to simpler proofs of Theorems 3, 6 and 7.